Sliding the top face
Glue a rubber cube to the floor and push its top face sideways with a force \(F\). The floor holds the bottom face, so the top slides over the bottom. The cube keeps its volume, but its square side faces become slanted. The shear stress is \(F/A\), where \(A\) is the area of the top face. The shear strain is the angle \(\theta\) through which the side faces turn, \(\theta \approx x/L\) for a shift \(x\) of a cube of height \(L\).
The shear modulus of a material, also called its modulus of rigidity, is the ratio of shearing stress to shear strain. Within the proportional limit, \(G = (F/A)/\theta\).
Move the slider to \(100\,\text{N}\). The top face has an area of \(0.01\,\text{m}^2\), so the shear stress is \(10^4\,\text{Pa}\). The shear angle is \(\theta = 10^4/(5 \times 10^5) = 0.02\,\text{rad}\), about one degree. The top shifts by \(x = L\theta = 0.1 \times 0.02\,\text{m} = 2\,\text{mm}\). Double the force, and the angle and the shift double too.
For metals the shear modulus is large: about \(8 \times 10^{10}\,\text{Pa}\) for steel and \(4 \times 10^{10}\,\text{Pa}\) for copper. For most solids \(G\) lies between a third and a half of \(Y\). The same force that shifts a rubber cube by \(2\,\text{mm}\) would shift a steel cube by about ten billionths of a metre.
A machine stands on a rubber pad \(2\,\text{cm}\) thick with a top area of \(0.04\,\text{m}^2\). The pad has \(G = 5 \times 10^5\,\text{Pa}\), and its bottom is fixed. The running machine pushes sideways on the pad with \(400\,\text{N}\). How far does the top of the pad move sideways?
The force acts along the top face, so this is shear. \(F = 400\,\text{N}\), \(A = 0.04\,\text{m}^2\), \(L = 0.02\,\text{m}\).
\(\theta = F/(GA)\), then \(x = L\theta\).
\(F/A = 10^4\,\text{Pa}\), so \(\theta = 10^4/(5 \times 10^5) = 0.02\,\text{rad}\), and \(x = 0.02 \times 0.02 = 4 \times 10^{-4}\,\text{m}\).
The top of the pad moves by \(0.4\,\text{mm}\). The thin pad shifts less than the tall cube at the same angle.
A rubber cube of side \(10\,\text{cm}\), with \(G = 5 \times 10^5\,\text{Pa}\), is glued to the floor. A force of \(250\,\text{N}\) acts along its top face. By how many millimetres does the top face shift?
A cube is sheared by a small tangential force on its top face. What happens to its volume?
Stretched longer, drawn thinner
Stretch a rubber band and watch its width: it becomes narrower as it becomes longer. A steel wire does the same, far too little to see. The fractional change in width, \(\Delta d/d\), is the lateral strain. It is negative when the longitudinal strain \(\varepsilon = \Delta L/L\) is positive.
Within the elastic range, the lateral strain is proportional to the longitudinal strain. Their ratio, with a minus sign to make it positive, is Poisson's ratio, written \(\sigma\). It has no unit. For steel it is about \(0.3\), and for most metals \(0.25\) to \(0.35\). Rubber is nearly \(0.5\). Cork is nearly zero, which is why a pushed cork does not swell and jam in a bottle.
Does the volume change?
A stretched wire grows longer but thinner. Whether its volume grows depends on which effect wins. The derivation below finds the answer for a round wire. One step is left blank for you to choose.
\[ V = \dfrac{\pi d^2}{4}\,L \]
Why. a round wire is a cylinder of diameter \(d\) and length \(L\).
\[ \dfrac{\Delta V}{V} = 2\,\dfrac{\Delta d}{d} + \dfrac{\Delta L}{L} \]
Why. for small changes of a product, the fractional changes add; \(d^2\) counts twice.
?
Your turn. put \(\Delta d/d = -\sigma\varepsilon\) and \(\Delta L/L = \varepsilon\). Which line is it?
\[ \dfrac{\Delta V}{V} = (1 - 2\sigma)\,\varepsilon \]
Read it. take \(\varepsilon\) out as a common factor.
For \(\sigma = 0.25\) and \(\varepsilon = 10^{-3}\), the volume grows by \((1 - 0.5) \times 10^{-3} = 5 \times 10^{-4}\), as the second chip of Fig 5.1 reads. For \(\sigma = 0.5\) the bracket is zero, and the volume does not change at all: the thinning exactly makes up for the lengthening. Rubber is close to this. No stable material has \(\sigma\) above \(0.5\), because then stretching would shrink it.
A steel wire \(1\,\text{m}\) long and \(1\,\text{mm}\) in diameter is stretched by \(0.5\,\text{mm}\). Poisson's ratio for steel is \(0.3\). Find the longitudinal strain, the lateral strain, the decrease in diameter and the fractional change in volume.
\(\varepsilon = 0.5 \times 10^{-3}/1 = 5 \times 10^{-4}\).
The lateral strain is \(-\sigma\varepsilon = -0.3 \times 5 \times 10^{-4} = -1.5 \times 10^{-4}\).
The diameter falls by \(1.5 \times 10^{-4} \times 1\,\text{mm} = 1.5 \times 10^{-4}\,\text{mm}\), about a seventh of a micrometre.
\(\Delta V/V = (1 - 0.6) \times 5 \times 10^{-4} = 2 \times 10^{-4}\). The volume grows a little.
A wire of a material with Poisson's ratio \(0.25\) is stretched to a longitudinal strain of \(2 \times 10^{-3}\). What is the fractional increase in its volume, in units of \(10^{-3}\)?
Three moduli, three kinds of change
Every modulus is a stress divided by the strain it produces. Young's modulus \(Y\) is tensile stress over longitudinal strain, and it governs a change of length. The bulk modulus \(B\) is the change of pressure over the volume strain, and it governs a change of volume. The shear modulus \(G\) is shear stress over shear angle, and it governs a change of shape.
So ask what the body is doing. A lift cable grows longer under its load: Young's modulus. The rocks of the sea-bed are squeezed from every side by the water above: bulk modulus. A shaft that turns a pump is twisted, so each thin layer of it slides past the next: shear modulus. A rivet cut across by two plates, and the air in a bicycle pump, are shear and bulk again.
A liquid at rest cannot resist a tangential force, however small. Its layers simply slide past one another and keep sliding, which is what flowing means. So a liquid has no shear modulus. It has no fixed length or shape to return to either, so it has no Young's modulus. But a liquid resists being squeezed into less room, so it has a bulk modulus. Gases, too, have only a bulk modulus.
Which of these are governed mainly by the shear modulus?
Why does water have a bulk modulus but no shear modulus and no Young's modulus?
A shear modulus needs a restoring force against a change of shape. Water at rest cannot resist any tangential force: its layers slide and keep sliding, so it flows, and there is no shear modulus. It has no fixed length or shape of its own to return to, so there is no Young's modulus either. But its molecules already touch, and squeezing them into less room is strongly resisted, so water has a large bulk modulus.
For a material that is the same in every direction, the moduli are not independent. Young's modulus, the shear modulus and Poisson's ratio are linked by \(Y = 2G(1 + \sigma)\). For steel, \(2 \times 8 \times 10^{10} \times 1.3 = 2.08 \times 10^{11}\,\text{Pa}\), close to the measured \(Y\). The link is proved in courses on the theory of elasticity.
In 1829 the French mathematician Siméon Denis Poisson published a theory of elastic solids built on forces between molecules. It predicted that a stretched rod should thin by exactly one quarter of its fractional stretch, for every material. Later measurements found other values: nearer one third for many metals, and nearly one half for rubber. His theory was too simple, but his ratio kept his name.
Look under the deck of an elevated Metro line where it rests on a pillar. The deck often sits on thick pads of rubber and steel plates, called elastomeric bearings. On a hot afternoon the long concrete deck expands by a few millimetres. The pads let it move by shearing, as in the sim above, instead of pushing hard on the pillars.
- Measuring the shear angle in degrees in \(G = F/(A\theta)\).\(\theta\) must be in radians, \(\theta \approx x/L\).
- Using the area of the side face in the shear stress.\(A\) is the area of the face the tangential force acts on, the top face.
- Forgetting the factor 2 in \(\Delta V/V = 2\,\Delta d/d + \Delta L/L\).The area goes as \(d^2\), so the lateral strain counts twice.
- Giving a liquid a shear modulus.A liquid flows under any tangential force; it has only a bulk modulus.